Calculus 1 Problems - I'm Stuck!!!!

[WhoopDDoos]

Hey guys, im in my first year of calculus and never really had a great class of mathematics growing up, so I took calculus 1 to see what I could do. I have these 2 problems to do and I am lost as last years easter egg. Any help would be very much appreciated :grin::grin::grin:


Problem 1:

A sector with central angle θ is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of θ such that the volume of the cone is a maximum.

Problem 2:

The side of a cube is measured with an error of at most 1.5%

a. What is the relative error of the volume if the sides of the cube are 2 inches?
b. If the volume of a cube with sides=2inches is expanding at the rate of 2% per second, calculate the rate at which the volume is increasing.
c. For the same expansion rate, calculate the rate of change of the surface area.

Thanks Guys, anything helps. :razz:

 

[Ishuda]

Hey guys, im in my first year of calculus and never really had a great class of mathematics growing up, so I took calculus 1 to see what I could do. I have these 2 problems to do and I am lost as last years easter egg. Any help would be very much appreciated :grin::grin::grin:


Problem 1:

A sector with central angle θ is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of θ such that the volume of the cone is a maximum.

Problem 2:

The side of a cube is measured with an error of at most 1.5%

a. What is the relative error of the volume if the sides of the cube are 2 inches?
b. If the volume of a cube with sides=2inches is expanding at the rate of 2% per second, calculate the rate at which the volume is increasing.
c. For the same expansion rate, calculate the rate of change of the surface area.

Thanks Guys, anything helps. :razz:

Although it isn't listed in the 'Read Before Posting' (from what I remember)
http://www.freemathhelp.com/forum/threads/41538-Read-Before-Posting!!
it is usually nice, IMO, to just pose one problem per post.

Anyway, first question: Since the equation for the volume of a cone involves the radius of the base of the cone and the height of the cone, we need to find these.

So, what is the radius? Draw a picture if you need to so that you can see that the circumference of the circle for the base of the cone would be the same as the arc length of the segment of the circle you cut out. What is the length of that segment and how is that circumference related to the radius of the circle for the base of the cone.

Now that we have the radius of the base of the cone, what is the height? Well, if we were to drop a perpendicular to the base plane from the top tip of the cone, that would be the height. That line is also the locus of the center of the circles in the planes parallel to the base plane. Thus, where it intersect the base plane is the center of the circle of the base. Now think of the triangle formed by the perpendicular to the base plane (call that side A), the line from that perpendicular to the circle on the base plane (call that side B), and then the line from there back up to the top tip of the cone (call that side C). We have the length of A is the height, the length of B is the radius of the base cone, and the length of C is what? Well the length of C is what compared to the original circle from which you cut your sector? Knowing B and C we can use the Pythagorean Theorem to compute the length of side A.

Now that we have the radius of the cone and its height, we can compute the volume.